The EU's product requirements process is documented in great detail on the European Commission website.
Under the principle of Mutual Recognition once a national agency has approved regulatory compliance no further checks on goods moving within the EU is required. This applies even when no EU Wide regulation exists, which is relevant for cars as per the example in the question;
Where no EU-wide specifications exist (about 15% of product rules are not harmonised in the EU), different specifications might apply in different EU countries. In such cases, you must only comply with the rules valid in your country. Other countries cannot forbid the sale of your product nor oblige you to modify it or do additional testing, provided you can prove that your products fulfil all the technical and quality requirements in your own country and offer a similar level of safety: this is the principle of mutual recognition.
This is not to say that spot checks or audits are never carried out, as happened during the 2013 Horsemeat scandal